Logic and Computer Design Fundamentals Chapter 2 Combinational

Logic and Computer Design Fundamentals Chapter 2 – Combinational Logic Circuits Part 2 – Circuit Optimization

Overview § Part 2 – Circuit Optimization 2 -4 Two-Level Circuit Optimization 2 -5 Map Manipulation Chapter 2 - Part 2 2

Circuit Optimization § Goal: To obtain the simplest implementation for a given function § Optimization is a more formal approach to simplification that is performed using a specific procedure or algorithm § Optimization requires a cost criterion to measure the simplicity of a circuit § Distinct cost criteria we will use: • Literal cost (L) • Gate input cost (G) • Gate input cost with NOTs (GN) Chapter 2 - Part 2 3

Literal Cost § Literal – a variable or it complement § Literal cost – the number of literal appearances in a Boolean expression corresponding to the logic circuit diagram § Examples: • F = AB+C(D+E) • F = AB+CD+CE • Which solution is best? L=5 L=6 Which is best? The same complexity? Chapter 2 - Part 2 4

Gate Input Cost § Gate input costs - the number of inputs to the gates in the implementation corresponding exactly to the given equation or equations. (G - inverters not counted, GN - inverters counted) § For SOP and POS equations, it can be found from the equation(s) by finding the sum of: • all literal appearances • the number of terms excluding single literal terms, (G) and • optionally, the number of distinct complemented single literals (GN). § Example: • Which solution is best? Chapter 2 - Part 2 5

Cost Criteria (continued) GN = G + 2 = 9 L=5 § F=A+BC+ BC G=L+2= 7 § Example 1: B C A F § L (literal count) counts the AND inputs and the single literal OR input. § G (gate input count) adds the remaining OR gate inputs § GN(gate input count with NOTs) adds the inverter inputs Chapter 2 - Part 2 6

Cost Criteria (continued) § Example 2: § § A B C F = A B C + AB C L = 6 G = 8 GN = 11 F = (A + C)( B + C)( A + B) L = 6 G = 9 GN = 12 F § Same function and same literal cost A § But first circuit has better B gate input count and better C gate input count with NOTs § Select it! => Gate input cost performs better. F Chapter 2 - Part 2 7

Boolean Function Optimization § Minimizing the gate input (or literal) cost of a (a set of) Boolean equation(s) reduces circuit cost. § We choose gate input cost. § Boolean Algebra and graphical techniques are tools to minimize cost criteria values. § Some important questions: • When do we stop trying to reduce the cost? • Do we know when we have a minimum cost? => Optimization problem § Treat optimum or near-optimum cost functions for two-level (SOP and POS) circuits first. § Introduce a graphical technique using Karnaugh maps (Kmaps, for short) Chapter 2 - Part 2 8

Karnaugh Maps (K-map) § A K-map is a collection of squares • • Each square represents a minterm The collection of squares is a graphical representation of a Boolean function Adjacent squares differ in the value of one variable Alternative algebraic expressions for the same function are derived by recognizing patterns of squares § The K-map can be viewed as • A reorganized version of the truth table Chapter 2 - Part 2 9

Some Uses of K-Maps § Provide a means for: • Finding optimum or near optimum § SOP and POS standard forms, and § two-level AND/OR and OR/AND circuit implementations for functions with small numbers of variables • Visualizing concepts related to manipulating Boolean expressions, and • Demonstrating concepts used by computeraided design programs to simplify large circuits Chapter 2 - Part 2 10

Two Variable Maps § A 2 -variable Karnaugh Map: • Note that all the adjacent minterms are differ in value of only one variable Chapter 2 - Part 2 11

From Truth Tables to K-Map § K-Maps can be used to simplify Boolean functions by systematic methods. Terms are selected to cover the 1’s cells in the map. § Example – Two variable function: Result of simplification : Chapter 2 - Part 2 12

K-Map Function Representation Example 2 -4: Result of simplification=? Chapter 2 - Part 2 13

K-Map Function Representation § Example: G(x, y) = x + y G = x+y y = 0 y = 1 x=0 0 1 x=1 1 1 § For G(x, y), two pairs of adjacent cells containing 1’s can be combined using the Minimization Theorem: G( x , y ) = (x y + x y )+ (xy + x y )= x + y Duplicate x y Chapter 2 - Part 2 14

Three Variable Maps § A three-variable K-map: § Where each minterm corresponds to the product terms: yz=00 x=0 x y z yz=01 yz=10 xyz xyz x y zin one y z an index x y z differs xyz § Note that if the binaryx=1 valuex for bit position, the minterms are adjacent on the K-Map Chapter 2 - Part 2 15

Three-Variable Maps § Reduced literal product terms for SOP standard forms correspond to rectangles on K-maps containing cell counts that are powers of 2. § Rectangles of 2 cells represent 2 adjacent minterms; of 4 cells represent 4 minterms that form a “pairwise adjacent” ring. • 2 cells rectangle will cancel 1 variable • 4 (=22 ) cells rectangle will cancel 2 variables • 8 (=23 ) cells rectangle will cancel 3 variables § What is the result for this case in three variables K-map? Chapter 2 - Part 2 16

Three Variable Maps § Example 2 -5: Simplify Result: Chapter 2 - Part 2 17

Three Variable Maps § Example 2 -6: Simplify Result: Chapter 2 - Part 2 18

Three Variable Maps § Example 2 -7: Simplify Result: Chapter 2 - Part 2 19

Three-Variable Map Simplification § Use a K-map to find an optimum SOP equation for Chapter 2 - Part 2 20

Four Variable Maps § Map and location of minterms: Chapter 2 - Part 2 21

Four Variable Terms § Four variable maps can have rectangles corresponding to: • A single 1 = 4 variables, (i. e. Minterm) • Two 1 s = 3 variables, • Four 1 s = 2 variables • Eight 1 s = 1 variable, • Sixteen 1 s = zero variables (i. e. Constant "1") Chapter 2 - Part 2 22

Four-Variable Maps § Example 2 -8: Simplify Result: Chapter 2 - Part 2 23

Four-Variable Maps § Example 2 -9: Simplify Result: Chapter 2 - Part 2 24

Four-Variable Map § F(W, X, Y, Z) = Sm(3, 4, 5, 7, 9, 13, 14, 15) Chapter 2 - Part 2 25

2 -5 Map Manipulation § Implicant: the rectangles on a map made up of squares containing 1 s correspond to implicants. § A Prime Implicant is a product term obtained by combining the maximum possible number of adjacent squares in the map into a rectangle with the number of squares a power of 2. § A prime implicant is called an Essential Prime Implicant if it is the only prime implicant that covers (includes) one or more minterms. § Prime Implicants and Essential Prime Implicants can be determined by inspection of a K-Map. § A set of prime implicants "covers all minterms" if, for each minterm of the function, at least one prime implicant in the set of prime implicants includes the minterm. Chapter 2 - Part 2 26

Example of Prime Implicants § Find ALL Prime Implicants CD C BD 1 1 BD A AB 1 1 1 1 D AD ESSENTIAL Prime Implicants C BD 1 1 1 BD 1 B A 1 1 1 1 D BC Minterms covered by single prime implicant Chapter 2 - Part 2 27

Another Example § Find all prime implicants for: • Hint: There are seven prime implicants! Chapter 2 - Part 2 28

Optimization Algorithm § Find all prime implicants. § Include all essential prime implicants in the solution § Select a minimum cost set of non-essential prime implicants to cover all minterms not yet covered. § Minimize the overlap among prime implicants as much as possible. In particular, in the final solution, make sure that each prime implicant selected includes at least one minterm not included in any other prime implicant selected. Chapter 2 - Part 2 29

Selection Rule Example § Simplify F(A, B, C, D) given on the KSelected Essential map. C 1 1 1 A 1 C 1 1 1 B 1 D 0 1 1 1 A 1 1 1 B 1 D Minterms covered by essential prime implicants Chapter 2 - Part 2 30

Example 2 -12 § Simplifying a function using the selection rule Result: Chapter 2 - Part 2 31

Product-of-Sums Optimization § The minterm not included in the function belong to the complement of the function. § Example 2 -13:

Don't Cares in K-Maps § Sometimes a function table or map contains entries for which it is known: • the input values for the minterm will never occur, or • The output value for the minterm is not used § In these cases, the output value need not be defined § Instead, the output value is defined as a “don't care” § By placing “don't cares” ( an “x” entry) in the function table or map, the cost of the logic circuit may be lowered. § Example 1: A logic function having the binary codes for the BCD digits as its inputs. Only the codes for 0 through 9 are used. The six codes, 1010 through 1111 never occur, so the output values for these codes are “x” to represent “don’t cares. ” Chapter 2 - Part 2 33

Don't Cares in K-Maps § Example 2 -14: Simplify Chapter 2 - Part 2 34

Example: BCD “ 5 or More” § The map below gives a function F 1(w, x, y, z) which is defined as "5 or more" over BCD inputs. With the don't cares used for the 6 non-BCD combinations: y F 1 (w, x, y, z) = w + x z + x y G = 7 00 01 03 02 w § This is much lower in cost than F 2 where 04 15 17 16 the “don't cares” were treated as "0 s. " x X 12 X 13 X 15 X 14 F 2(w, x, y, z) = w x z + w x y G = 12 § For this particular function, cost G for the 1 8 1 9 X 11 X 10 POS solution for F 1(w, x, y, z) is not changed z by using the don't cares. Chapter 2 - Part 2 35